Geometry Point, Line, and Plane Postulates

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In this next lesson, we are going to go over some postulates that have to do with points, lines, and planes.0002

First, let's talk about postulates: what is a postulate?0011

A postulate is a statement that is assumed to be true; this is also called an axiom.0015

Postulates are accepted as fact without having to be proved.0023

Theorems are statements that have to be proved; you have to prove that it is true.0029

But postulates--we can just use them without any question if it is true or not--we don't have to prove it at all; it is just true.0035

And some postulates in your textbook--you might see that they are titled 2-2 or Postulate 2-1 or something.0046

Remember: when you name a postulate, you don't name it by that number that is used in your book,0057

because different books use different numbers, and it is in a different order.0062

If it doesn't have a name--if it just has a number, like Postulate 2.2, then remember that you have to write out the whole thing.0067

You can't just call it by the number that your book uses.0076

The first postulate that we are going to go over: Through any two points, there is exactly one line.0084

If there are any two points--I can draw two points however I want--maybe two points like that.0094

Through any two points, I can only draw one line through those two points, like that.0101

And there is no way that I can draw any other line.0110

So, if I have another two points, there is only one line that can be drawn through those two points.0113

The next one: Through any three points not on the same line, there is exactly one plane.0123

Through any three points not on the same line--meaning that they are not collinear, like that, there is exactly one plane.0130

I can only draw one plane that covers those three points--I can't draw any other plane.0140

Just like this one, through any two points, I can only draw one line--I can't draw any other type of line0148

that is going to go through those same two points--it is the same thing here.0154

Through any three points, I can only draw one single plane that is going to cover those points.0158

A line contains at least two points--"at least" meaning infinite--it contains two and a lot more.0166

So, a line contains at least two points.0178

A plane (remember, the fourth one--the next one) contains at least three points not on the same line.0189

If I have a plane, then this plane is going to contain at least three points; it is actually many, many, many--0204

but at least three points not on the same line, because if they are collinear, then it is just going to be on the same line.0213

But if they are not collinear, then it is going to be on the same plane; so this plane contains at least three points not on the same line.0223

The next one: If two points lie in a plane, then the entire line containing those two points lies in that plane.0233

So again, if two points lie in a plane (let me draw a plane, and two points lying in that plane),0244

then the entire line containing those two points lies in that plane.0256

The line that I can draw through those two points is going to also be in that plane.0262

So, if I have two points in a plane, then the line (remember: you can only draw one line through those points)0272

that you can draw is also going to be in that plane.0280

If two lines intersect, then they intersect in exactly one point.0288

If I have two lines, where they intersect is right here; where they intersect is going to be one point.0293

There is no way that they could intersect in any more than one point, because lines, we know, go straight.0304

Now, if we can bend it, then maybe it can come back around and meet again.0313

But we know that lines can't do that; it just goes straight, so their intersection is always going to be one point.0319

If two planes intersect, then their intersection is a line; so if I have (now, I am a very bad draw-er, but say I have) a plane like this,0330

and then I have a plane like this, so this is where they are intersecting; then this, where they intersect,0345

right here--that place where they are touching, where they are meeting, is a line.0361

When two lines intersect, it is going to be a point; when two planes intersect, it is going to be a line.0372

We can't just say that these two planes are going to intersect at a point,0378

because then that is not true; it is not just a single point--it is all of this right here.0384

So, it is going to be a line.0388

OK, if you want to review over the postulates again, just go ahead and rewind, or just go back and go over them again.0396

We are going to use the postulates to do a few example problems.0406

Using the postulates, determine if each statement is true or false.0411

Points A, B, and E...first of all, let's actually go over this diagram.0416

We have a plane: this is plane N; this is point A, right here; this is point B; this is C, point D; this is plane N--0423

this plane is N, right there; this point is I; this is point E.0437

So, points A, B, and E line in plane N; points A (this is point A), B, and point E (that is point E, right there) lie in plane N.0446

And we know that this is false, because E does not lie in it; A lies in it; B lies in the plane; but E does not, so this is false.0463

The next one: Points A, B, C, and E are coplanar.0480

"Coplanar" means that they are on the same plane.0488

Well, A, B...look at this...C, and E are coplanar; now, they might not be on plane N all together,0492

but they actually are coplanar, because this point...2, 3...and this one right here...I can form a plane0506

that is going to contain these four points, so this right here is true.0523

They are coplanar; it is not plane N, but they can lie on some plane, a different plane.0532

BC does not lie in plane N: here is BC right here; BC does not lie in plane N.0542

Well, it does actually lie in plane N, so this one is false, because B and C both lie (this point, B, and this point, C, lie) in plane N.0553

Remember the postulate where it says that, if two points are in the plane, then the line containing those two points also lies in the plane.0570

So, since point B and point C lie in the plane, BC has to lie in the plane.0580

Points A, B, and D are collinear: are they collinear?0589

They are coplanar, because they are all on plane N; but they are not collinear,0597

because, for it to be collinear, they have to be on the same line; and A, B, and D are not, so this one is false.0601

OK, let's go over a few more: now, we are going to determine if these are true or false.0617

Just to go over this diagram again: this right here is plane R; this right here is plane P; these are all the points.0628

Points B, D, A are part of this plane, and then, it is also part of this plane.0641

E is on plane P; H and I are not on either of them.0647

Points A, B, and D lie in plane R: is that true?0656

Here is plane R; B lies in it, D, and A; yes, it is true.0665

Points B, D, E, and F are coplanar; B, D, E, and F...well, B, D, and E are coplanar,0674

and B, D, and F are coplanar; but all four of them together--they are not coplanar.0694

So, there is no way that we can draw those four points on the same plane, so this one is false.0701

BA lies in plane P; BA, this segment right here, lies in P.0711

Well, I know that point B lies in plane P; point A lies in plane P; so the line containing those two points also has to be on that plane.0723

So, this one is true.0736

OK, we are going to go over a few more examples.0741

Use "always," "sometimes," or "never" to make each statement a true statement.0747

Intersecting lines are [always, sometimes, or never] coplanar.0752

If we have intersecting lines, no matter how we draw them (we can have them like this, or maybe like this),0760

intersecting lines are actually always going to be coplanar.0777

Can you draw a plane that contains those two points? Yes, so this one is always--always coplanar.0780

They are always going to be on the same plane.0790

Two planes [always/sometimes/never] intersect in exactly one point.0798

So, again, let me try to draw this out; I have a plane, and I have another plane...something like that.0804

Do they intersect? They intersect right here; when they intersect, are they intersecting at one point?0824

No, they intersect at a line; so this one is never: two planes never intersect in exactly one point.0834

It is always going to be a line.0848

Three points are [always, sometimes, or never] coplanar.0858

Well, if I have three points, are they going to be coplanar?0865

Yes, they are always going to be coplanar, because no matter how I draw these three points, I can always draw a plane around them.0878

Whether it is like that, or whether it is like they are collinear--they are going to be coplanar, the three points.0891

The next one: A plane containing two points of a line [always, sometimes, or never] contains the entire line.0904

A plane containing two points of a line contains the entire line--this is always.0917

As long as the two points are in that plane, the line has to also be in that plane.0934

Four points are [always, sometimes, or never] coplanar.0943

Well, this is actually going to be...let's see...if I have a plane like this, say I draw a line through that plane;0949

I can have point, point...if I have points A, B, C, and then right here, D, are all four points coplanar?0966

Now, what if I have this point right here, E? E is on this plane.0983

So, in this case, A, B, C, and E are coplanar; but A, B, C, and D are not coplanar; so this would be sometimes.0988

Two lines [always/sometimes/never] meet in more than one point.1006

Two lines, when they intersect...do they always meet at one point? Sometimes? Or never?1013

This is always at one point; can they meet in more than one point? No, so this one is never.1022

They can never meet in more than one point; they always have to meet in one point.1032

That is it for this lesson; thank you for watching Educator.com!1040